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volume 4, issue 4, article 70, 2003.

Received 15 September, 2003;

accepted 17 September, 2003.

Communicated by:Th. M. Rassias

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Journal of Inequalities in Pure and Applied Mathematics

SOME INEQUALITIES ASSOCIATED WITH A LINEAR OPERATOR DEFINED FOR A CLASS OF MULTIVALENT FUNCTIONS

V. RAVICHANDRAN, N. SEENIVASAGAN AND H.M. SRIVASTAVA

Department of Computer Applications Sri Venkateswara College of Engineering Sriperumbudur 602105, Tamil Nadu, India.

EMail:vravi@svce.ac.in Department of Mathematics Sindhi College

123 P.H. Road, Numbal

Chennai 600077, Tamil Nadu, India.

EMail:vasagan2000@yahoo.co.in Department of Mathematics and Statistics University of Victoria

Victoria, British Columbia V8W 3P4 Canada.

EMail:harimsri@math.uvic.ca

URL:http://www.math.uvic.ca/faculty/harimsri/

c

2000Victoria University ISSN (electronic): 1443-5756 122-03

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Some Inequalities Associated with a Linear Operator Defined

for a Class of Multivalent Functions

V. Ravichandran, N. Seenivasagan and

H.M. Srivastava

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Abstract

The authors derive several inequalities associated with differential subordina- tions between analytic functions and a linear operator defined for a certain family ofp-valent functions, which is introduced here by means of this linear operator. Some special cases and consequences of the main results are also considered.

2000 Mathematics Subject Classification:Primary 30C45.

Key words: Analytic functions, Univalent and multivalent functions, Differential sub- ordination, Schwarz function, Ruscheweyh derivatives, Hadamard prod- uct (or convolution), Linear operator, Convex functions, Starlike func- tions.

The present investigation was supported, in part, by the Natural Sciences and Engi- neering Research Council of Canada under Grant OGP0007353.

Contents

1 Introduction, Definitions and Preliminaries . . . 3 2 Inequalities Involving the Linear OperatorLp(a, c) . . . 11 3 Further Results Involving Differential Subordination Between

Analytic Functions . . . 16 References

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Some Inequalities Associated with a Linear Operator Defined

for a Class of Multivalent Functions

V. Ravichandran, N. Seenivasagan and

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1. Introduction, Definitions and Preliminaries

LetA(p, n)denote the class of functionsf normalized by (1.1) f(z) = zp+

X

k=p+n

akzk (p, n∈N:={1,2,3, . . .}), which are analytic in the open unit disk

U:={z :z ∈C and |z|<1}. In particular, we set

A(p,1) =:Ap and A(1,1) =:A =A1.

A function f∈ A(p, n) is said to be in the class A(p, n;α) if it satisfies the following inequality:

(1.2) R

1 + zf00(z) f0(z)

< α (z ∈U;α > p).

We also denote by K(α) and S(α), respectively, the usual subclasses of A consisting of functions which are convex of order α inUand starlike of order αinU. Thus we have (see, for details, [3] and [9])

(1.3) K(α) :=

f:f ∈ A andR

1 + zf00(z) f0(z)

> α (z ∈U; 05α <1)

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Some Inequalities Associated with a Linear Operator Defined

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and

(1.4) S(α) :=

f:f ∈ AandR

zf0(z) f(z)

> α (z∈U; 05α <1)

. In particular, we write

K(0) =:K and S(0) =:S.

For the above-defined classA(p, n;α)ofp-valent functions, Owa et al. [5]

proved the following results.

Theorem A. (Owa et al. [5, p. 8, Theorem 1]). If f(z)∈ A(p, n;α)

p < α5p+1 2n

,

then

(1.5) R

f(z) zf0(z)

> 2p+n

(2α+n)p (z ∈U).

Theorem B. (Owa et al. [5, p. 10, Theorem 2]). If f(z)∈ A(p, n;α)

p < α5p+1 2n

,

then

(1.6) 0<R

zf0(z) f(z)

< (2α+n)p

2p+n (z ∈U).

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Some Inequalities Associated with a Linear Operator Defined

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V. Ravichandran, N. Seenivasagan and

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In fact, as already observed by Owa et al. [5, p. 10], various further special cases of (for example) TheoremBwhenp=n = 1were considered earlier by Nunokawa [4], Saitoh et al. [7], and Singh and Singh [8].

The main object of this paper is to present an extension of each of the inequalities (1.5) and (1.6) asserted by Theorem A and Theorem B, respec- tively, to hold true for a linear operator associated with a certain general class A(p, n;a, c, α)ofp-valent functions, which we introduce here.

For two functionsf(z)given by (1.1) andg(z)given by g(z) =zp +

X

k=p+n

bk zk (p, n∈N),

the Hadamard product (or convolution)(f∗g) (z)is defined, as usual, by (1.7) (f ∗g) (z) :=zp+

X

k=p+n

akbkzk =: (g∗f) (z).

In terms of the Pochhammer symbol(λ)kor the shifted factorial, since (1)k=k! (k ∈N0 :=N∪ {0}),

given by

(λ)0 := 1 and (λ)k :=λ(λ+ 1)· · ·(λ+k−1) (k ∈N), we now define the functionφp(a, c;z)by

(1.8) φp(a, c;z) := zp+

X

k=1

(a)k (c)k zk+p

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Some Inequalities Associated with a Linear Operator Defined

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V. Ravichandran, N. Seenivasagan and

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z ∈U; a ∈R; c∈R\Z0; Z0 :={0,−1,−2, . . .}

.

Corresponding to the functionφp(a, c;z), Saitoh [6] introduced a linear opera- torLp(a, c)which is defined by means of the following Hadamard product (or convolution):

(1.9) Lp(a, c)f(z) := φp(a, c;z)∗f(z) (f ∈ Ap) or, equivalently, by

(1.10) Lp(a, c)f(z) :=zp+

X

k=1

(a)k

(c)k ak+p zk+p (z ∈U).

The definition (1.9) or (1.10) of the linear operator Lp(a, c) is motivated essentially by the familiar Carlson-Shaffer operator

L(a, c) := L1(a, c),

which has been used widely on such spaces of analytic and univalent functions in U asK(α) and S(α)defined by (1.3) and (1.4), respectively (see, for ex- ample, [9]). A linear operatorLp(a, c), analogous toLp(a, c)considered here, was investigated recently by Liu and Srivastava [2] on the space of meromorphi- callyp-valent functions in U. We remark in passing that a much more general convolution operator than the operatorLp(a, c)considered here, involving the generalized hypergeometric function in the defining Hadamard product (or con- volution), was introduced earlier by Dziok and Srivastava [1].

Making use of the linear operatorLp(a, c)defined by (1.9) or (1.10), we say that a functionf ∈ A(p, n)is in the aforementioned general classA(p, n;a, c, α)

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Some Inequalities Associated with a Linear Operator Defined

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if it satisfies the following inequality:

(1.11) R

Lp(a+ 2, c)f(z) Lp(a+ 1, c)f(z)

< α z ∈U; α >1; a∈R; c∈R\Z0

.

The Ruscheweyh derivative off(z)of orderδ+p−1is defined by (1.12) Dδ+p−1 f(z) := zp

(1−z)δ+p ∗f(z) (f∈ A(p, n) ; δ ∈R\(−∞,−p]) or, equivalently, by

(1.13) Dδ+p−1 f(z) :=zp +

X

k=p+n

δ+k−1 k−p

ak zk

(f ∈ A(p, n) ; δ ∈R\(−∞,−p]).

In particular, if δ = l(l+p∈N), we find from the definition (1.12) or (1.13) that

(1.14) Dl+p−1 f(z) = zp (l+p−1)!

dl+p−1 dzl+p−1

zl−1 f(z) , (f ∈ A(p, n) ; l+p∈N).

Since

(1.15) Lp(δ+p,1)f(z) =Dδ+p−1 f(z),

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Some Inequalities Associated with a Linear Operator Defined

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V. Ravichandran, N. Seenivasagan and

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(f ∈ A(p, n) ; δ ∈R\(−∞,−p]),

which can easily be verified by comparing the definitions (1.10) and (1.13), we may set

(1.16) A(p, n;δ+p,1, α) =:A(p, n;δ, α).

Thus a function f ∈ A(p, n)is in the classA(p, n;δ, α)if it satisfies the fol- lowing inequality:

(1.17) R

Dδ+p+1 f(z) Dδ+p f(z)

< α, (z ∈U; α >1; δ∈R\(−∞,−p]).

Finally, for two functions f and g analytic in U, we say that the function f(z)is subordinate tog(z)inU, and write

f ≺g or f(z)≺g(z) (z ∈U), if there exists a Schwarz functionw(z), analytic inUwith

w(0) = 0 and |w(z)|<1 (z ∈U), such that

(1.18) f(z) =g w(z)

(z ∈U).

In particular, if the function g is univalent in U, the above subordination is equivalent to

f(0) =g(0) and f(U)⊂g(U).

In our present investigation of the above-defined general classA(p, n;a, c, α), we shall require each of the following lemmas.

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Some Inequalities Associated with a Linear Operator Defined

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V. Ravichandran, N. Seenivasagan and

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Lemma 1. (cf. Miller and Mocanu [3, p. 35, Theorem 2.3i (i)]). Let Ω be a set in the complex plane Cand suppose that Φ (u, v;z)is a complex-valued mapping:

Φ :C2×U→C, where

u=u1+iu2 and v =v1+iv2.

Also letΦ (iu2, v1;z)∈/ Ω for allz ∈Uand for all realu2 andv1 such that

(1.19) v1 5−1

2n 1 +u22 . If

q(z) = 1 +cnzn+cn+1zn+1+· · · is analytic inUand

Φ (q(z), zq0(z) ;z)∈Ω (z ∈U), then

R{q(z)}>0 (z ∈U).

Lemma 2. (cf. Miller and Mocanu [3, p. 132, Theorem 3.4h]). Letψ(z)be univalent inUand suppose that the functionsϑandϕare analytic in a domain D ⊃ ψ(U) with ϕ(ζ) 6= 0 when ζ ∈ ψ(U). Define the functions Q(z) and h(z)by

(1.20) Q(z) := zψ0(z)ϕ ψ(z)

and h(z) :=ϑ ψ(z)

+Q(z),

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Some Inequalities Associated with a Linear Operator Defined

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V. Ravichandran, N. Seenivasagan and

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and assume that

(i) Q(z)is starlike univalent inU and

(ii) R

zh0(z) Q(z)

>0 (z ∈U). If

(1.21) ϑ

q(z)

+zq0(z)ϕ q(z)

≺h(z) (z ∈U), then

q(z)≺ψ(z) (z ∈U) andψ(z)is the best dominant.

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Some Inequalities Associated with a Linear Operator Defined

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2. Inequalities Involving the Linear Operator L

p

(a, c)

By appealing to Lemma 1 of the preceding section, we first prove Theorem1 below.

Theorem 1. Let the parametersaandαsatisfy the following inequalities:

(2.1) a >−1 and 1< α51 + n 2(a+ 1). Iff(z)∈ A(p, n;a, c, α), then

(2.2) R

Lp(a, c)f(z) Lp(a+ 1, c)f(z)

> 2a+n

2α(a+ 1)−2 +n (z∈U) and

(2.3) R

Lp(a+ 1, c)f(z) Lp(a, c)f(z)

< 2α(a+ 1)−2 +n

2a+n (z ∈U).

Proof. Define the functionq(z)by

(2.4) (1−β)q(z) +β = Lp(a, c)f(z)

Lp(a+ 1, c)f(z) (z∈U), where

(2.5) β := 2a+n

2α(a+ 1)−2 +n.

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Some Inequalities Associated with a Linear Operator Defined

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Then, clearly,q(z)is analytic inUand

q(z) = 1 +cn zn+cn+1 zn+1+· · · (z ∈U).

By a simple computation, we observe from (2.4) that (2.6) (1−β)zq0(z)

(1−β)q(z) +β = z Lp(a, c)f(z)0

Lp(a, c)f(z) − z Lp(a+ 1, c)f(z)0 Lp(a+ 1, c)f(z) . Making use of the familiar identity:

(2.7) z Lp(a, c)f(z)0

=aLp(a+ 1, c)f(z)−(a−p)Lp(a, c)f(z), we find from (2.6) that

(1−β)zq0(z)

(1−β)q(z) +β = 1 +a Lp(a+ 1, c)f(z)

Lp(a, c)f(z) −(a+ 1) Lp(a+ 2, c)f(z) Lp(a+ 1, c)f(z), which, in view of (2.4), yields

Lp(a+ 2, c)f(z)

Lp(a+ 1, c)f(z) = 1

a+ 1 + 1 a+ 1

a

(1−β)q(z) +β − (1−β)zq0(z) (1−β)q(z) +β

or, equivalently,

(2.8) Lp(a+ 2, c)f(z)

Lp(a+ 1, c)f(z) = 1 a+ 1

1 + a−(1−β)zq0(z) (1−β)q(z) +β

.

If we defineΦ(u, v;z)by

(2.9) Φ(u, v;z) := 1

a+ 1

1 + a−(1−β)v (1−β)u+β

,

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Some Inequalities Associated with a Linear Operator Defined

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then, by the hypothesis of Theorem1thatf ∈ A(p, n;a, c, α), we have R{Φ (q(z), zq0(z);z)}=R

Lp(a+ 2, c)f(z) Lp(a+ 1, c)f(z)

< α (z ∈U; α >1).

We will now show that

R{Φ (iu2, v1;z)}=α

for all z ∈ U and for all real u2 and v1 constrained by the inequality (1.19).

Indeed we find from (2.9) that R{Φ (iu2, v1;z)}= 1

a+ 1

1 +R

a−(1−β)v1 (1−β)iu2

= 1

a+ 1

1 +R

[a−(1−β)v1][β−(1−β)iu2] (1−β)2u222

= 1

a+ 1

1 + [a−(1−β)v1]β (1−β)2u222

,

so that, by using (1.19), we have (2.10) R{Φ (iu2, v1;z)}= 1

a+ 1

1 + β[a+12n(1−β)(1 +u22)]

(1−β)2u222

(z∈U). From the inequalities in (2.1), we get

n 2β2 =

a+1

2n(1−β)

(1−β),

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Some Inequalities Associated with a Linear Operator Defined

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and hence the function

a+ 12n(1−β)(1 +x2) (1−β)2x22

is an increasing function forx=0. Thus we find from (2.10) that R{Φ (iu2, v1;z)}= 1

a+ 1

1 + a+ 12n(1−β) β

=α (z ∈U). The first assertion (2.2) of Theorem1follows by applying Lemma1.

Next, we define the functionψ(z)by ψ(z) := Lp(a, c)f(z)

Lp(a+ 1, c)f(z) (z ∈U),

where β is given by (2.5). Then, in view of the already proven assertion (2.2) of Theorem1, we have

(2.11) R{ψ(z)}> β > 0 (z ∈U) so that

(2.12) R

1 ψ(z)

>0 (z ∈U).

Since (2.12) holds true, we have R{ψ(z)}R

1 ψ(z)

5|ψ(z)| · 1

|ψ(z)| = 1,

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Some Inequalities Associated with a Linear Operator Defined

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or

R 1

ψ(z)

5 1

R{ψ(z)} (z ∈U), which, in view of (2.11), yields

0<R 1

ψ(z)

< 1

β (z ∈U) which is the second assertion (2.3) of Theorem1.

The following result is a special case of Theorem1obtained by taking a=δ+p and c= 1.

Corollary 1. If

f(z)∈ A(p, n;δ, α)

δ+p > 1; 15α <1 + n 2(δ+p+ 1)

,

then

R

Dδ+p−1f(z) Dδ+pf(z)

> 2δ+ 2p+n

2α(δ+p+ 1)−2 +n (z ∈U), and

R

Dδ+pf(z) Dδ+p−1f(z)

< 2α(δ+p+ 1)−2 +n

2δ+ 2p+n (z ∈U).

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Some Inequalities Associated with a Linear Operator Defined

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3. Further Results Involving Differential Subordi- nation Between Analytic Functions

We begin by proving the following result.

Lemma 3. Let the functionsq(z)andψ(z)be analytic inUand suppose that ψ(z)6= 0 (z ∈U)

is also univalent inUand that0(z)/ψ(z)is starlike univalent inU. If

(3.1) R

α β

1 ψ(z)+

1 + zψ00(z)

ψ0(z) −zψ0(z) ψ(z)

>0, (z ∈U; α, β ∈C; β 6= 0)

and

(3.2) α

q(z)−β zq0(z) q(z) ≺ α

ψ(z) −β zψ0(z) ψ(z) , (z ∈U; α, β ∈C; β 6= 0), then

q(z)≺ψ(z) (z ∈U) andq(z)is the best dominant.

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Some Inequalities Associated with a Linear Operator Defined

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Proof. By setting

ϑ(ζ) = α

ζ and ϕ(ζ) = −β ζ,

it is easily observed that bothϑ(ζ)andϕ(ζ)are analytic inC\{0}and that ϕ(ζ)6= 0 (ζ ∈C\ {0}).

Also, by letting

(3.3) Q(z) =zψ0(z)ϕ ψ(z)

=−β zψ0(z) ψ(z)

and

(3.4) h(z) =ϑ ψ(z)

+Q(z) = α

ψ(z) −β zψ0(z) ψ(z) ,

we find thatQ(z)is starlike univalent inUand that R

zh0(z) Q(z)

=R α

β 1 ψ(z)+

1 + zψ00(z)

ψ0(z) −zψ0(z) ψ(z)

>0, (z ∈U; α, β ∈C; β 6= 0),

by the hypothesis (3.1) of Lemma3. Thus, by applying Lemma2, our proof of Lemma3is completed.

We now prove the following result involving differential subordination be- tween analytic functions.

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Theorem 2. Let the functionψ(z)6= 0 (z ∈U)be analytic and univalent inU and suppose that0(z)/ψ(z)is starlike univalent inUand

(3.5) R

a ψ(z)+

1 + zψ00(z)

ψ0(z) −zψ0(z) ψ(z)

>0 (z ∈U; a ∈C\ {−1}).

Iff ∈ Apsatisfies the following subordination:

(3.6) Lp(a+ 2, c)f(z)

Lp(a+ 1, c)f(z) ≺ 1 a+ 1

1 + a−zψ0(z) ψ(z)

(z∈U), then

(3.7) Lp(a, c)f(z)

Lp(a+ 1, c)f(z) ≺ψ(z) (z ∈U) andψ(z)is the best dominant.

Proof. Let the functionq(z)be defined by q(z) := Lp(a, c)f(z)

Lp(a+ 1, c)f(z) (z ∈U; f ∈ Ap), so that, by a straightforward computation, we have

(3.8) zq0(z)

q(z) = z Lp(a, c)f(z)0

Lp(a, c)f(z) − z Lp(a+ 1, c)f(z)0 Lp(a+ 1, c)f(z) ,

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which follows also from (2.6) in the special case whenβ = 0.

Making use of the familiar identity (2.7) once again (or directly from (2.8) withβ = 0), we find that

Lp(a+ 2, c)f(z)

Lp(a+ 1, c)f(z) =a Lp(a+ 1, c)f(z)

Lp(a, c)f(z) −(a+ 1) Lp(a+ 2, c)f(z) Lp(a+ 1, c)f(z) + 1

= 1

a+ 1

1 + a

q(z)− zq0(z) q(z)

,

which, in light of the hypothesis (3.6) of Theorem2, yields the following sub- ordination:

a

q(z) − zq0(z) q(z) ≺ a

ψ(z) − zψ0(z)

ψ(z) (z ∈U).

The assertion (3.7) of Theorem2now follows from Lemma3.

Remark 1. If the functionψ(z)is such that

R{ψ(z)}>0 (z ∈U)

and if0(z)/ψ(z) is starlike in U, then the condition (3.5) is satisfied for a >0.

In its special case when

a=δ+p and c= 1, Theorem2yields the following result.

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Corollary 2. Let the functionψ(z)6= 0 (z ∈U)be analytic and univalent inU and suppose that0(z)/ψ(z)is starlike univalent inUand

R

δ+p ψ(z) +

1 + zψ00(z)

ψ0(z) − zψ0(z) ψ(z)

>0 (z ∈U; δ∈R\(−∞, p]).

Iff ∈ Asatisfies the following subordination:

Dδ+p+1f(z)

Dδ+pf(z) ≺ 1 δ+p+ 1

1 + δ+p−zψ0(z) ψ(z)

(z ∈U), then

Dδ+p−1f(z)

Dδ+pf(z) ≺ψ(z) (z ∈U).

Lastly, by using a similar technique as above, we can prove Theorem3be- low.

Theorem 3. Iff ∈ A(p, n)and (3.9) 1 + zf00(z)

f0(z) ≺p 1 +Bzn

1 +Azn − n(A−B)zn (1 +Azn)(1 +Bzn), (z ∈U; −15B < A51),

then

(3.10) pf(z)

zf0(z) ≺ 1 +Azn

1 +Bzn (z ∈U).

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Some Inequalities Associated with a Linear Operator Defined

for a Class of Multivalent Functions

V. Ravichandran, N. Seenivasagan and

H.M. Srivastava

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Proof. Let the functionq(z)be defined by

(3.11) q(z) := pf(z)

zf0(z) (z∈U; f ∈ A(p, n)), so that

(3.12) 1 + zf00(z)

f0(z) = p

q(z) −zq0(z) q(z) .

If the functionψ(z)is defined by ψ(z) := 1 +Azn

1 +Bzn (−15B < A51; z ∈U), then, in view of (3.9) and (3.12), we get

p

q(z) − zq0(z) q(z) ≺ p

ψ(z) − zψ0(z)

ψ(z) (z ∈U).

The result (Theorem3) now follows from Lemma3(withα =pand β = 1).

The following result is a simple consequence of Theorem3.

Corollary 3. Iff ∈ Asatisfies the following subordination:

1 + zf00(z)

f0(z) ≺ 1−4z+z2

1−z2 (z ∈U),

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Some Inequalities Associated with a Linear Operator Defined

for a Class of Multivalent Functions

V. Ravichandran, N. Seenivasagan and

H.M. Srivastava

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then

(3.13) R

f(z) zf0(z)

>0 (z ∈U) or, equivalently,f is starlike inU(that is,f ∈ S).

Remark 2. The foregoing analysis can be applied mutatis mutandis in order to rederive TheoremAof Owa et al. [5]. Indeed, if

(3.14) f(z)∈ A(p, n;α)

p < α5p+1 2n

,

then we can first show that

1 + zf00(z)

f0(z) ≺ψ(z) (z ∈U), where

ψ(z) :=p 1 +Bzn

1 +Azn − n(A−B)zn

(1 +Azn)(1 +Bzn) = p(1 +Bzn)2−n(A+ 1)zn (1 +Azn)(1−zn)

A= 1−2β; B =−1; β= 2p+n 2α+n

.

By letting

u(θ) :=R{ψ(z)} z =eiθ/n∈∂U; 05θ52nπ ,

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Some Inequalities Associated with a Linear Operator Defined

for a Class of Multivalent Functions

V. Ravichandran, N. Seenivasagan and

H.M. Srivastava

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it is easily seen for

u(θ) = (1−A) [2p+n(1 +A)−2pcosθ]

2(1 +A2+ 2Acosθ) (05θ52nπ) that

(3.15) u(θ)=u(π) =(1−A) [2p+n(1 +A) + 2p]

2(1−A)2 =α (05θ52nπ), which shows thatq(U)contains the half-planeR(w)5α, whereq(z)is given, as before, by(3.11). Thus, under the hypothesis(3.14), we have the subordina- tion(3.9)and hence(by Theorem3)also the subordination(3.10), which leads us to the assertion(1.5)of TheoremA.

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Some Inequalities Associated with a Linear Operator Defined

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References

[1] J. DZIOK AND H.M. SRIVASTAVA, Classes of analytic functions associ- ated with the generalized hypergeometric function, Appl. Math. Comput., 103 (1999), 1–13.

[2] J.-L. LIU ANDH.M. SRIVASTAVA, A linear operator and associated fam- ilies of meromorphically multivalent functions, J. Math. Anal. Appl., 259 (2001), 566–581.

[3] S.S. MILLER AND P.T. MOCANU, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Ap- plied Mathematics (No. 225), Marcel Dekker, New York and Basel, 2000.

[4] M. NUNOKAWA, A sufficient condition for univalence and starlikeness, Proc. Japan Acad. Ser. A Math. Sci., 65 (1989), 163–164.

[5] S. OWA, M. NUNOKAWA AND H.M. SRIVASTAVA, A certain class of multivalent functions, Appl. Math. Lett., 10 (2) (1997), 7–10.

[6] H. SAITOH, A linear operator and its applications of first order differential subordinations, Math. Japon., 44 (1996), 31–38.

[7] H. SAITOH, M. NUNOKAWA, S. FUKUI AND S. OWA, A remark on close-to-convex and starlike functions, Bull. Soc. Roy. Sci. Liège, 57 (1988), 137–141.

[8] R. SINGH AND S. SINGH, Some sufficient conditions for univalence and starlikeness, Colloq. Math., 47 (1982), 309–314.

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Some Inequalities Associated with a Linear Operator Defined

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H.M. Srivastava

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[9] H.M. SRIVASTAVA AND S. OWA (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992.

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